Let $\theta \not\in \mathbb{Q}$. We know that $(n\theta)_{n \geq 1}$ is equidistributed modulo 1.

Let $\epsilon_n = \mathrm{sign}\bigl(\sin(n\pi \theta)\bigr)$ and $S_N= \sum_{n=1}^N \epsilon_n$.

I'm looking for a "good" asymptotic bound for $|S_N|$ (not $|S_N|\leq N$ obviously).

It looks like for any $x>0$, we should have $S_N =o(n^x)$, or even better, that $(S_N)$ is bounded, but is it?